5 Best O' Math Psych Books

The Wall Street Journal has gotten in the habit of publishing a list of the 5 books that are the best for whatever subject. I am going to give my thoughts on the five best (i.e., classic and mandatory) books for the mathematical psychologist. The thing that makes many of the articles in these volumes amazing compared to reading Frege's Begriffsschrift or the Principia (actually, either of the two Principias- Newton's or Whitehead & Russell's) is that you can still learn from the science contained within, they are not merely of historical or philosophical interest. I am not saying that the context of the aforementioned texts are wrong, but if you want to learn logic you pick up a modern logic textbook and similarly for physics. (Though I think that mere historical and philosophical interest is more often that not more than sufficient reason to study something, which is why I own copies of Newton's Principia and W&R's up to *56.) Perhaps, later on, I will do the same for some related fields such as mathematical logic, philosophical logic, and game theory.

Mandatory books for the mathematical psychologist who specializes in applications of logic and computation to cognition (i.e., this list is naturally prejudiced toward my research interests) and in no particular order:

1. Anderson, J. & Rosenfield, E. (Eds.) (1999). Neurocomputing: Foundations of Research. Cambridge: MIT Press.

This has many of the classic articles in the field, to include the most classic of them all, viz. McCulloch & Pitts (1943)- the seminal paper on neural networks, Rosenblatt (1958) that introduces the perceptron, Minsky & Papert (1969) where they express the limitations of the perceptron, Hopfield (1982) where he brought neural nets to the popular scientific front and combined many disparate ideas elegantly, and many more- 43 papers (or excerpts from books) in all.

2. Shannon, C.E. & McCarthy, J. (1956). "Automata Studies," Annals of Mathematical Studies, Number 34. Princton: Princeton University Press.

An early collection of works in automata from the incomparable Annals of Mathematical Studies from Princeton University. The papers come in three topics: Finite Automata, Turing Machines, and Synthesis of Automata. The first section has exceptional papers on logic, neural nets, and automata by Kleene, von Neumann, Minsky, and Moore. These (along with the McCulloch & Pitts paper) are the founding papers on finite automata. The next two sections also have splendid papers by Davis, de Leeuwe, et al, MacKay, and a pair of papers by Uttley.

3. Luce, R.D., Bush, R.R., Galanter, E. (1963). Handbook of Mathematical Psychology, Vol. II, Chapters 9-14. New York: John Wiley & Sons.

There are six exceptional papers in this volume, where any one would be worth purchasing the book. The first two papers are on stochastic learning theory (Sternberg) and stimulus sampling theory (Atkinson & Estes) while the last paper is on mathematical models of social interaction (Rapoport). All are great introductions to their respective fields and still relevant. However, in the middle there is a trinity of papers by Chomsky: Chomsky & Miller's "Introduction to the formal analysis of natural languages," Chomsky's "Formal properties of grammars," and Miller's & Chomsky's "Finitary Models of Language Users." I cannot adequately express how wonderful these papers are, whatever you feel about Chomsky's politics (for good or ill), just ignore all of that and enjoy the pure genius of he and Miller combining Chomsky's groundbreaking work in linguistics in the 1950s with automata theory. So freakin' brilliant. These papers are on par with Frege's Foundations of Arithmetic. Really. They are that good.

4. Minsky, M.L. (1967). Computation: Finite and Infinite Machines. Englewood Cliffs: Prentice-Hall.

One of the first (if not the first) textbooks that took the ideas that are scattered in the pre-1965 articles in (1) and the articles in (2) and synthesized them clearly using contemporary logical notation. Still a lot to learn from the book. His proof of the equivalence of neural nets and finite automata is still one of my favorite proofs.

5. Krantz, Luce, Suppes, Tversky (1971). Foundations of Measurement, Vol I: Additive and Polynomial Representations. Mineola: Dover.

While the second of the three volumes, which is on Geometrical, Threshold, and Probabilistic Relations, is more clearly linked to my research, the first volume is essential to read. It is essential since it lays down the foundations of what measurement theory is. Measurement theorists examine how the physical world (both nature and humans) can be measured quantitatively.

Bonus. Luce, R.D. (1964). "The Mathematics Used in Mathematical Psychology," The American Mathematical Monthly, Vol. 71, No. 4.

Since this is an article, not a book, I have included it as a bonus; this article explains the position of mathematics and psychology and lays out the challenges that are still facing the mathematical psychologist today. It also includes a nice synopsis of measurement theory. The moral of the story: mathematical psychologists are getting a 400 year late (if not a three thousand year late) start than mathematical physicists and dealing with a much more complex subject, viz. humans, and so new mathematical techniques will have to be invented in order to solve some of the open problems in mathematical psychology.